The JacksonRichmond 4CT Constant is EXACTLY 10/27
By Shalosh B. Ekhad and Doron Zeilberger
.pdf
.tex
Written: Feb. 7, 2024
In their recent claimed computerfree proof of the Four Color Theorem, David Jackson
and Bruce Richmond attempted to use sophisticated "asymptotic analysis" to explicitly compute a certain number whose positivity (according to them) implies this famous theorem. While the jury is still out
whether their valiant attempt holds water, we prove, in this modest note, that this constant equals
exactly 10/27. We also point out that their evaluation of this constant must be erroneous, for two
good reasons. Finally, as an encore, we state many similar, but more complicated, results.
Maple package
Jackmond.txt , a Maple package to find explicit expressions, as rational functions of n, of the
ratio of the coefficient of x^n in
(Sum(f(i)*x^i,i=1..infinity))^{r} divided by f(n)
if
Sum(f(i)*x^i,i=1..infinity)
is an algebraic formal power series.
Sample input an output for Jackmond.txt

If you want to explicit expressions as rational functions of n, of the ratio of the coefficient of x^n of (Sum(2*(4*n+1)!/((n+1)!*(3*n+2)!)*x^n,n=1..infinity)^{r} divided by 2*(4*n+1)!/((n+1)!*(3*n+2)!
for 2 ≤ r ≤ 11
the input file yields the output file.

If you want to go all the way to r=18,
the input file yields the output file.
Personal Journal of Shalosh B. Ekhad and Doron Zeilberger